The ED developed here deals with the coupled evolution of a quantum scalar field together with a classical dynamical background. The class of theories that allow such evolving geometries are often called instances of geometrodynamics. In a typical geometrodynamics (see e.g., [1]}[2]}), the primary object of interest is the evolving three-metric \(g_{ij}(x)\) , whose dynamics must be suitably constrained so that the time evolution sweeps a four-dimensional space-time with metric \(^{4}g_{\mu \nu }\)  (\(\mu ,\nu ,...\)  \(=0,1,2,3\) ). Thus, despite the intrinsic dynamics at play, it is nonetheless appropriate to make reference to the enveloping space-time, if only as formal scaffolding that can be later removed.Remarkably, as discussed by Teitelboim [3]}, although certain aspects of the formalism, indeed, rely crucially on space-time itself, such notions are ultimately absent from the operative equations of geometrodynamics. In view of this, we can rightfully regard three-dimensional space as taking a primary role, with space-time being taken as a secondary construct.
